I know they're power chords but my guitar teacher was explaining that all power chords are derived from "actual chords" so I need to know what "actual chords" the power chords in the song are derived from.
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mmmm. well, powerchords technically aren't chords, just harmonies. a proper chord has 3 different notes (not octaves) according to some guys, but 2 note chords can be accepted by some too.

I dont know the song, but i'll just explain you: (im to lazy to use the code thingy)
the notes are
D|-2| E
A|-2| B
E|-0| E

this means you are playing an E chord. The E chord is formed by E (1st, or root note) then it's 3rd (G) and it's fifth(B). Powerchords are 1st and 5ths, so they are called fifth chords by some guys.

The 3rd defines if a chord is major or minor (minor if there are 3 half steps between 1st and 3rd and major if there are 4 half steps). There is no third in powerchords, so they are not major nor minor (they actually are, according to the key, but let your teacher explain that because it's more complex)

some "powerchords" with 3rd's are like this:

D|-5-| G
A|-7-| E
E|-0-| E
this is the first chord in Orion. It has the root note (e) and it's third (g). there are 3 half steps between E and G (F, F#, G), so it's E minor.. However, there are only two notes so they could be considered a minor 3rd harmony instead of a chord.

I hope that explains something. sorry for sh*t english

I believe its E phrygian for the main riff and the prechorus seems to be in E aeolian.

This would make the chords:


...for the verse riff and...


...for the pre-chorus (i.e. giving it a dominant sound), finally, the intro and chorus sections are:

Em7 to Dmaj7

If that helps?

I've worked out all the chords up to 7th chords, but it would sound odd to play metallica with Jazz chords...
I would expect he is asking you to check out whether the power chord you play would be major or minor if a 3rd is added. Best way to check that out is to try it - when playing with the song try and make it major or minor and see which sounds right. The first E chord sounds fine as an Em but sounds a bit dodgy as an E major for example. Now try and work out the rest by playing through with the song replacing power chords with full chords.
Ask your teacher how to figure it out.

Basically, it'll have the same root note but there'll be one or two more notes than the power chord - however, you need to have some understanding of keys and harmony before that information is useful to you. Hence, ask your teacher, it's his job!